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This document will describe how to represent masses and inertias in
mechanics and use of the RigidBody and Particle classes.

It is assumed that the reader is familiar with the basics of these topics, such
as finding the center of mass for a system of particles, how to manipulate an
inertia tensor, and the definition of a particle and rigid body. Any advanced
dynamics text can provide a reference for these details.

The mass is specified exactly as is in a particle. Similar to the
Particle‘s .point, the RigidBody‘s center of mass, .masscenter
must be specified. The reference frame is stored in an analogous fashion and
holds information about the body’s orientation and angular velocity. Finally,
the inertia for a rigid body needs to be specified about a point. In
mechanics, you are allowed to specify any point for this. The most
common is the center of mass, as shown in the above code. If a point is selected
which is not the center of mass, ensure that the position between the point and
the center of mass has been defined. The inertia is specified as a tuple of length
two with the first entry being a Dyadic and the second entry being a
Point of which the inertia dyadic is defined about.

In mechanics, dyadics are used to represent inertia ([Kane1985],
[WikiDyadics], [WikiDyadicProducts]). A dyadic is a linear polynomial of
component unit dyadics, similar to a vector being a linear polynomial of
component unit vectors. A dyadic is the outer product between two vectors which
returns a new quantity representing the juxtaposition of these two vectors. For
example:

Where \(\mathbf{\hat{a}_x}\mathbf{\hat{a}_x}\) and
\(\mathbf{\hat{a}_x}\mathbf{\hat{a}_y}\) are the outer products obtained by
multiplying the left side as a column vector by the right side as a row vector.
Note that the order is significant.

A vector in a reference frame can be represented as
\(\begin{bmatrix}a\\b\\c\end{bmatrix}\) or \(a \mathbf{\hat{i}} + b
\mathbf{\hat{j}} + c \mathbf{\hat{k}}\). Similarly, a dyadic can be represented
in tensor form:

Just as with vectors, the later representation makes it possible to keep track
of which frames the dyadic is defined with respect to. Also, the two
components of each term in the dyadic need not be in the same frame. The
following is valid: